| L(s) = 1 | + 0.347·3-s + 4-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 16-s − 19-s + 25-s − 0.652·27-s − 0.879·36-s + 0.532·39-s + 1.53·47-s + 0.347·48-s + 49-s + 1.53·52-s − 53-s − 0.347·57-s − 1.87·59-s + 64-s − 1.87·73-s + 0.347·75-s − 76-s − 1.87·79-s + 0.652·81-s − 89-s + 100-s + 1.53·101-s + ⋯ |
| L(s) = 1 | + 0.347·3-s + 4-s − 0.879·9-s + 0.347·12-s + 1.53·13-s + 16-s − 19-s + 25-s − 0.652·27-s − 0.879·36-s + 0.532·39-s + 1.53·47-s + 0.347·48-s + 49-s + 1.53·52-s − 53-s − 0.347·57-s − 1.87·59-s + 64-s − 1.87·73-s + 0.347·75-s − 76-s − 1.87·79-s + 0.652·81-s − 89-s + 100-s + 1.53·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.629944499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629944499\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2003 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - 0.347T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.53T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + 1.87T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 + 1.87T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.064888061779371384429558299190, −8.607462817657275460934427201340, −7.85305608860504719413333730021, −6.95700635433192104564140022369, −6.13648691027400823583051673066, −5.68031559858507047814753067409, −4.30350435254199879302278573543, −3.30057283543544462659792325523, −2.58262598939111800571373723922, −1.43042549484316165441871962923,
1.43042549484316165441871962923, 2.58262598939111800571373723922, 3.30057283543544462659792325523, 4.30350435254199879302278573543, 5.68031559858507047814753067409, 6.13648691027400823583051673066, 6.95700635433192104564140022369, 7.85305608860504719413333730021, 8.607462817657275460934427201340, 9.064888061779371384429558299190
